The boundary of the gravitational Standard-Model Extension

A modification of General Relativity that is based on the gravitational Standard-Model Extension and incorporates nondynamical background fields has recently been studied via the ADM formalism. Our objective in this paper is to develop a better understanding of the additional contributions that arise on the spacetime boundary $\partial M$. An extension of the previously introduced boundary terms, which are relevant in the context of asymptotically flat spacetimes, follows from the decomposition of $\partial M$ into timelike and spacelike hypersurfaces. Furthermore, we present an alternative method of deriving the field equations satisfied by the induced metric on the purely spacelike hypersurfaces of the foliated spacetime. This leads to the dynamical part of the Einstein equations modified by the background fields. Our results have the potential to be applicable in various contexts such as modified black holes and cosmology.